Analysis of RHC for stabilization of nonautonomous parabolic equations under uncertainty

TL;DR

This paper investigates the stabilization of nonautonomous parabolic equations with uncertain parameters using Receding Horizon Control, analyzing how actuator number influences stability and failure probability under uncertainty.

Contribution

It introduces a framework for stabilizing time-varying parabolic equations with uncertain data via RHC, considering random fields and actuator configurations.

Findings

Expected stabilizability depends on the number of actuators.

An upper bound for failure probability related to actuator count and parameters.

Analysis covers both uniform and log-normal distributions of diffusion coefficients.

Abstract

Stabilization of a class of time-varying parabolic equations with uncertain input data using Receding Horizon Control (RHC) is investigated. The diffusion coefficient and the initial function are prescribed as random fields. We consider both cases, uniform and log-normal distributions of the diffusion coefficient. The controls are chosen to be finite-dimensional and enter into the system as a linear combination of finitely many indicator functions (actuators) supported in open subsets of the spatial domain. Under suitable regularity assumptions, we study the expected (averaged) stabilizability of the RHC-controlled system with respect to the number of actuators. An upper bound is also obtained for the failure probability of RHC in relation to the choice of the number of actuators and parameters in the equation.